Notes on su$(1,2)\oplus$u$(1)$ Chern-Simons theory and Torsional Newton-Cartan gravity

Abstract

In this study, we investigate three-dimensional torsional Newton-Cartan (TNC) gravity by gauging the su$(1,2)\oplus$u$(1)$ algebra and construct its action using the Chern-Simons theory. This TNC exhibits novel features, including the fact that the gauge fields associated with both dilatation and rotation symmetries transform non-trivially under Galilean boosts. This theory also reproduces the Schrödinger gravity acquired by gauging the extended $z=2$ Schrödinger algebra arXiv:1604.08054 via a large speed of light ($1/c$)-expansion. In particular, we explain that the $z=2$ Lifshitz vacuum solution appearing in Schrödinger gravity is related to the null reduction of 4d $\Omega$-background up to a conformal factor. Based on these results, we revisit the identification between the extended Schrödinger algebra and bosonic analogue of super BMS algebra arXiv:1905.13154. We interpret that this relation originates from the ${\mathcal{W}}_3^{(2)}$ algebra which acts as the bosonic analogue of $\mathcal{N}=2$ superconformal algebra.

arXiv comment45 pages, 2 figures; v2, minor corrections in staements, typos corrected, reference added; v3, ournal version, add section about excitation solution, comparison with TTNC, and references added

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